# Chapter 3 Hypothetical Syllogisms

Hypothetical Syllogisms

As stated in Chapter 2, a hypothetical syllogism is a syllogism with at least one conditional premise, that is, at least one ―if…then…‖ premise. The ―If…then…‖ relationship may be expressed in ordinary language by using a number of different terms. In checking hypothetical syllogisms expressed in ordinary language for their validity, it is therefore useful to be able to translate such conditional vocabulary into standard conditional form.

The Vocabulary of Conditional Statements

A conditional statement can be said to be in standard conditional form when it is stated with the antecedent stated first and the consequent stated second, and when the antecedent is preceded by the term ―If‖ and the consequent by the term ―then.‖ For example, the following statement is in standard conditional form: If it rains then the roof will leak

In contrast, the following statement is not in standard conditional form: The roof will leaks if it rains

The following table provides some common conditional vocabulary and illustrates how to translate it into standard conditional form:

Conditional

Vocabulary

Only if

Example

Standard Conditional Form

It rains only if there are clouds.

Only if there are clouds does it rain.

If it rains then there are clouds

1

If and only if

You are eligible to vote if and only if

you have a valid voter registration

card

If you have a valid voter registration

card then you are eligible to vote

&

If you are eligible to vote then you

have a valid voter registration card

On condition

that

I will go on condition that you pay

me.

If I you pay me then I will go.

Entails,

Bachelorhood entails being

unmarried.

If one is a bachelor then one is

unmarried

Implies

Bachelorhood implies being

unmarried

In the event that

In the event that there’s a fire, take

the stairs.

If there is a fire then take the stairs

When

When he lies, his left eye twitches.

If he lies then his left eye twitches

Where

Where there’s smoke, there’s fire.

If there’s smoke then there’s fire

Assuming that,

Assuming that weather permits, the

graduation will be outdoors.

If the weather permits then

graduation will be outdoors

Given that

Given that the weather permits, the

graduation will be outdoors.

Provided that

The bill will become law provided

that the President signs it.

If the president signs the bill then the

bill will become law

Provided that the president signs it,

the bill will become law.

Notice that ―only if‖ does not mean the same thing as ―if.‖ In asserting that ―p only if q‖ I am asserting that you cannot have p without q, which translates into standard conditional form as ―If p then q‖. Consider our example, ―It rains only if there are clouds.‖ This means that it cannot rain unless there are clouds, which translates into standard conditional form as ―If it rains then there are clouds.‖1

1

Notice also that if q is a necessary condition of p then p is also a sufficient condition of q. That is, if you can’t have p without q, then if you have p you will also have q.

2

In translating ―only if‖ statements into standard conditional form, a useful rule is that whatever immediately follows the “only if” becomes the consequent. So, in the statement, ―Only if there are clouds does it rain,‖ the consequent is ―there are clouds‖— since this is what immediately follows the ―only if.‖

Notice that the term ―if and only if‖ yields two conditional statements, each of which reverses the order of the other’s antecedent and consequent. This is because this term says two things: It says ―if‖ and it says ―only if‖. Thus ―p if and only if q‖ means ―p if q,‖ and ―p only if q.‖ And since p only if q is translated as ―if p then q‖ the result is two conditionals in reverse order: ―if p then q‖ and ―if q then p‖. For example, ―You are eligible to vote if and only if you have a valid voter registration...

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